(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 760416, 14805]*) (*NotebookOutlinePosition[ 761065, 14827]*) (* CellTagsIndexPosition[ 761021, 14823]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Computer Project #1", "Section"], Cell["Ben Sauerwine", "Text"], Cell[CellGroupData[{ Cell["Operator and Algorithm Definitions", "Subsection"], Cell["I have selected the infinity norm for all norms. ", "Text"], Cell[BoxData[{ \(\(A[i_, \ j_, \ n_, \ \[Alpha]_]\ := \ \ If[i \[Equal] j, \ If[Or[i \[Equal] 1, \ i \[Equal] n], \ 1, \ \[Alpha]], \ If[Or[i \[Equal] j + 1, i \[Equal] j - 1], \ \(-1\), \ 0]];\)\), "\[IndentingNewLine]", \(MultiplyA[x_, \ n_, \ \[Alpha]_]\ := \ Table[\[Sum]\+\(j = 1\)\%n A[i, \ j, \ n, \[Alpha]]\ x[\([j]\)\ ], \ {i, \ 1, \ n}]\), "\[IndentingNewLine]", \(\(InitialX[k_, \ n_]\ := \ Table[Cos[k\ \[Pi]\ \((i\ - \ 1\/2)\)\/\((n - \ 1\/2)\)], \ {i, \ 1, \ n}];\)\), "\[IndentingNewLine]", \(\(InfinityNorm[x_]\ := \ Max[Abs[x]];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(RichardsonStep[x_, \ b_, \ n_, \ \[Alpha]_]\ := \ Block[{r\ }, \ r\ = \ Table[ b[\([i]\)] - \ \[Sum]\+\(j = 1\)\%n A[i, \ j, \ n, \ \[Alpha]]\ x[\([j]\)], \ {i, \ 1, \ n}]; \ x\ + \ r];\)\), "\[IndentingNewLine]", \(\(JacobiStep[x_, \ b_, \ n_, \ \[Alpha]_]\ := \ \ Table[\(b[\([i]\)] - \ \[Sum]\+\(j = \ 1\)\%n If[Not[j \[Equal] i], A[i, \ j, \ n, \ \[Alpha]]\ x[\([j]\)], \ 0]\)\/A[i, \ i, \ \ n, \ \[Alpha]], \ {i, \ 1, \ n}];\)\), "\[IndentingNewLine]", \(\(GaussSeidlStep[x_, \ b_, \ n_, \ \[Alpha]_]\ := \ \ Block[{u\ = \ x}, For[i = 1, i \[LessEqual] n, \ \(i++\), \ u[\([i]\)]\ = \ \((b[\([i]\)]\ - \ \[Sum]\+\(j = 1\)\%n If[ Not[j \[Equal] i], A[i, \ j, \ n, \ \[Alpha]]\ u[\([j]\)], 0])\)/ A[i, \ i, \ n, \ \[Alpha]]]; \ u];\)\), "\[IndentingNewLine]", \(\(KaczmarzStep[x_, \ b_, \ n_, \ \[Alpha]_]\ := \ \ Block[{u\ = \ x}, For[i = 1, i \[LessEqual] n, \ \(i++\), \ u[\([i]\)]\ = u[\([i]\)]\ + \ \((\ \[Sum]\+\(k = 1\)\%n\((A[k, \ i, \ n, \ \[Alpha]] \((b[\([k]\)]\ - \ \[Sum]\+\(j = \ 1\)\%\(k\ - \ 1\)A[k, \ j, \ n, \ \[Alpha]]\ u[\([j]\)] - \ \ \[Sum]\+\(j = i\)\%n A[k, \ j, \ n, \ \[Alpha]]\ u[\([j]\)])\))\))\)/\((\ \[Sum]\+\(j = 1\)\%n\((A[i, \ j, \ n, \ \[Alpha]] A[i, \ j, \ n, \ \[Alpha]])\))\)]; \ u];\)\), "\[IndentingNewLine]", \(\(RunAlgorithm[algorithm_, \ steps_, \ x_, \ b_, \ n_, \ \[Alpha]_]\ := \ Block[{result, \ a, \ curr\ = \ x, \ residualnorms, \ errnorms}, result = {0, \ 0, \ 0}; \ errnorms = Table[0, \ {i, \ 1, steps}]; \ residualnorms = \ Table[0, \ {i, \ 0, \ steps}]\ ; \ For[a = 1, \ a \[LessEqual] \ steps, \ \(a++\), \ errnorms[\([a]\)]\ = \ InfinityNorm[curr]; \ residualnorms[\([a]\)]\ = \ InfinityNorm[MultiplyA[curr, \ n, \ \[Alpha]]\ - \ b]; \ curr\ = \ N[algorithm[curr, \ b, \ n, \ \[Alpha]]]]; \ result[\([1]\)]\ = \ curr; \ result[\([2]\)]\ = \ errnorms; \ result[\([3]\)]\ = \ residualnorms; \ result];\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Experiment 1: Kaczmarz, k = 0, n = 64, vary \[Alpha]\ \>", "Subsection"], Cell["\<\ Take the Kaczmarz algorithm, use k = 0 for the initial estimate, n = 64. Use \ 50 steps, and try \[Alpha]=3, 2, 1.99, 1.95, 1.9, 1.5.\ \>", "Text"], Cell[BoxData[{ \(\(exp11\ = \ N[RunAlgorithm[KaczmarzStep, 50, \ InitialX[0, \ 64], \ Table[0, \ {i, \ 1, 64}], \ 64, \ 3]];\)\), "\[IndentingNewLine]", \(\(exp12\ = \ N[RunAlgorithm[KaczmarzStep, 50, \ InitialX[0, \ 64], \ Table[0, \ {i, \ 1, 64}], \ 64, \ 2]];\)\), "\[IndentingNewLine]", \(\(exp13\ = \ N[RunAlgorithm[KaczmarzStep, 50, \ InitialX[0, \ 64], \ Table[0, \ {i, \ 1, 64}], \ 64, \ 1.99]];\)\), "\[IndentingNewLine]", \(\(exp14\ = \ N[RunAlgorithm[KaczmarzStep, 50, \ InitialX[0, \ 64], \ Table[0, \ {i, \ 1, 64}], \ 64, \ 1.95]];\)\), "\[IndentingNewLine]", \(\(exp15\ = \ N[RunAlgorithm[KaczmarzStep, 50, \ InitialX[0, \ 64], \ Table[0, \ {i, \ 1, 64}], \ 64, \ 1.9]];\)\), "\[IndentingNewLine]", \(\(exp16\ = \ N[RunAlgorithm[KaczmarzStep, 50, \ InitialX[0, \ 64], \ Table[0, \ {i, \ 1, 64}], \ 64, \ 1.5]];\)\)}], "Input"], Cell["\<\ In all cases above, the algorithm converges to zero. 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exp12[\([2]\)], \ {PlotRange \[Rule] {{0, \ 8}, \ {0, \ 1}}, PlotJoined \[Rule] True, PlotStyle \[Rule] Hue[0.2]}];\)\), "\[IndentingNewLine]", \(\(PlotExp13\ = \ ListPlot[ exp13[\([2]\)], \ {PlotRange \[Rule] {{0, \ 8}, \ {0, \ 1}}, PlotJoined \[Rule] True, PlotStyle \[Rule] Hue[0.3]}];\)\), "\[IndentingNewLine]", \(\(PlotExp14\ = \ ListPlot[ exp14[\([2]\)], \ {PlotRange \[Rule] {{0, \ 8}, \ {0, \ 1}}, PlotJoined \[Rule] True, PlotStyle \[Rule] Hue[0.4]}];\)\), "\[IndentingNewLine]", \(\(PlotExp15\ = \ ListPlot[ exp15[\([2]\)], \ {PlotRange \[Rule] {{0, \ 8}, \ {0, \ 1}}, PlotJoined \[Rule] True, PlotStyle \[Rule] Hue[0.5]}];\)\), "\[IndentingNewLine]", \(\(PlotExp16\ = \ ListPlot[ exp16[\([2]\)], \ {PlotRange \[Rule] {{0, \ 8}, \ {0, \ 1}}, PlotJoined \[Rule] True, PlotStyle \[Rule] Hue[0.6]}];\)\), "\[IndentingNewLine]", \(Show[{PlotExp11, \ PlotExp12, \ PlotExp13, \ PlotExp14, \ PlotExp15, \ PlotExp16}]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell["\<\ Kaczmarz' error norms 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